Question

The procedure adopted in the Gauss- Jordan method in solving linear simultaneous equation is that
A. It is required to assume initial approximate values of the variables
B. It reduces the given system of equation to an equivalent triangular system
C. It reduces the given system of equation to a diagonal matrix
D. The given matrix is factored into lower and upper triangular matrix
Choose the correct answer

• A. A, B and D only
• B. A only
• C. B and C only
• D. C only

Hint:

Remember the difference: Gaussian Elimination reduces the matrix to a triangular form, whereas Gauss-Jordan reduces it to a diagonal (or identity) form, eliminating the need for back-substitution.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The Gauss-Jordan method is an extension of Gaussian elimination used to solve systems of linear simultaneous equations.
It relies on performing elementary row operations on the augmented matrix \([A|B]\).

Step 2: Detailed Explanation:
Let us evaluate each procedure listed:
- Statement A: Assuming initial approximate values is a feature of iterative numerical methods, such as the Gauss-Seidel or Jacobi methods, not direct elimination methods. Thus, A is incorrect.
- Statement B: Reducing the system to an equivalent upper triangular system is the standard procedure of Gaussian elimination (followed by back-substitution to find the variables). Thus, B is incorrect.
- Statement C: In the Gauss-Jordan method, row operations are applied both below and above the pivot elements.
This reduces the coefficient matrix \(A\) directly to a diagonal matrix (or identity matrix), allowing the values of the variables to be read directly without back-substitution. Thus, C is correct.
- Statement D: Factoring the matrix into lower and upper triangular matrices is the procedure used in LU Decomposition (or Crout's/Doolittle's methods). Thus, D is incorrect.
Consequently, only Statement C describes the Gauss-Jordan method.

Step 3: Final Answer:
The correct option is C only.